Structured screens for controlled spreading of light

ABSTRACT

Structured screens for the controlled spreading, diffusion, or scattering of an incident beam are provided. The screens are composed of microstructures ( 1,2 ) whose configurations and distribution on the surfaces of the screen are precisely determined. In certain embodiments, the configurations and/or their distribution is randomized. The structured screens can be used as diffusing screens or display screens.

I. FIELD OF THE INVENTION

[0001] The present invention discloses an optical device composed of asubstrate whose surface or surfaces, which may be flat or curved,contain a distribution of microstructures capable of spreading (alsoreferred to herein as “diffusing” or “scattering”) an incident beam overa controlled range of angles and with controllable intensity variationacross the useful field. The said device is referred to herein as a“structured screen” or simply a “screen.”

[0002] The structured screens of the invention have a plurality ofapplications which generally fall into two major categories, namely,diffusive screens and display screens. More particularly, applicationsof the structured screens include, but are not limited to,homogenization of illumination, in conjunction with or as a focusingscreen for photographic cameras, to provide uniform illumination overspecific viewing angles, and applications in back-lit displays, liquidcrystal flat panel displays, and other types of displays either as astandalone device or in conjunction with other necessary hardware.

[0003] Examples of instruments where the structured screens can be usedinclude, but are not limited to, photographic cameras, computer screens,television sets, projection screens, cellular phones, and general imagedisplay equipment.

II. BACKGROUND OF THE INVENTION

[0004] A. Diffusive Screens

[0005] In many applications there is the need for devices whose purposeis to spread an illumination beam over a certain field of interest witha desired intensity variation. Such devices are generally referred to inthe art as diffusive screens, diffusion plates, or diffusers.

[0006] In its simplest version, a diffusive screen is made of a roughsurface with a relief pattern that can be typically described byGaussian statistics. To fabricate such diffusive screens several methodshave been proposed. Among these one can distinguish three basiccategories.

[0007] First, there are diffusive screens based on a random surfacestructure (ground glass). Such diffusive screens are commerciallyavailable at low cost. However, because there is little control overtheir diffusing characteristics, the performance of such screens is verylimited and only of interest in applications with very flexible andloose requirements.

[0008] A second class of diffusive screens is obtained by holographicrecording of a speckle pattern. This class offers more flexibility thanground glass screens in tailoring the diffusion pattern. However, suchholographic diffusive screens tend to generate images with a grainyappearance, which may be unpleasant for viewing purposes. Also, thesudden intensity variations associated with speckles lead to non-uniformillumination over restricted viewing angles.

[0009] The third class of diffusive screens include those where acertain substrate has its surface modified according to some reliefpattern. An example includes arrays of microlenses which provide lightdiffusion. This third class offers better control of the relief patternthan either ground glass screens or holographic screens.

[0010] There has been considerable effort to address the problem oflight diffusion as briefly summarized by the following U.S. patents.

[0011] U.S. Pat. No. 4,427,265 discloses a diffusive screen with anirregular arrangement of curved surfaces superposed on a periodicmicrolens array. The goal is to maintain the light diffusing propertieswhile avoiding some of the artifacts associated with the underlyingperiodic array. The curvature of each microlens is controlled onaverage.

[0012] U.S. Pat. No. 5,733,710 describes various arrangements ofmicrolenses generated by mask exposure with microlens location beingvaried through mask rotation. It also discloses the combination of adiffusive screen structure and a Fresnel lens on opposite sides of thesame substrate.

[0013] U.S. Pat. No. 4,826,292 discloses a diffusion plate with a reliefstructure composed of cones created by ion bombardment and etching.

[0014] U.S. Pat. No. 5,871,653 discloses fabrication methods to obtain adiffusive screen structure based on microlens arrays for use in flatpanel displays.

[0015] Some of the issues that must be addressed when designing adiffusive screen include controllable viewing angles, controllableintensity variation over the useful viewing field, resolution, absenceof visual artifacts, and efficient use of the incident illumination. Toachieve full control of design capabilities and obtain the best possiblediffusing performance for a given application one must be able tocontrol the surface-relief pattern with adequate precision.

[0016] The relief control achieved in the prior art is limited to simplearrangements where individual structures might have some curvature oroptical power. In particular, existing art in the fabrication ofmicrolens arrays includes, among others, the techniques disclosed inU.S. Pat. Nos. 5,871,653, 5,536,455, 5,324,623, and 5,300,263. Currentmethods are based on polymer melting, thermal relaxation, ion exchangediffusion, surface tension effects, and etch smoothing. These methodsoffer little control over the microlens shape, except that it is nearlyspherical.

[0017] The quality obtained in the prior art is a largely statisticalprocess because there has been no strict control of the positioningand/or shaping of the structures used to achieve diffusion. Theelementary structures that compose the arrays are conventionally nearlyspherical shapes. As often found in the patent literature, theelementary structures that compose a diffusive screen are looselydescribed as “curved” simply because there is little control over theirshape. For other types of relief structures not described by curvedmicrolenses, the surface is obtained by random means of only statisticalcontrol, such as surface bombardment.

[0018] It is therefore clear that there exists a need forwell-controlled diffusing surfaces with elementary microstructures thatare well defined and chosen to meet specific diffusion requirements.

[0019] B. Display Screens

[0020] Applications that involve the display of information requireappropriate means of delivery to allow the user some form of interactionwith the information, be it access to a database or simply watching amovie. Such systems are usually composed of (1) a light engine whichprovides illumination, (2) optics to transfer the optical information,and (3) a display screen which provides the immediate delivery of thevisual information to the user. The light engine and optics are, for allpractical purposes, invisible to the user.

[0021] The display screen, however, represents the element of directcontact with the user and, for this reason, needs to embody in the bestpossible way the performance of the system. In other words, the displayscreen provides the immediate impression to the user and the quality ofthe image it can provide determines, to a great extent, the acceptanceor not of a particular system.

[0022] Some of the issues relevant to the performance of display screensare efficiency (brightness), resolution (ability to resolve features andavoid aliasing effects), gain (scattering over specified angular range),low speckle (graininess of image associated with random structures ofsome screen surface designs), contrast (clear distinction betweencolors), and ambient light rejection (screen looks black when lightengine is turned off). These are just some of the issues that must betaken into account in the design of the light engine, optics, anddisplay screen, since these all work together.

[0023] Traditionally, the approaches used to design display screens havebeen the same as those used for diffusing screens. Thus, display screenshave incorporated random elements in the screen surface without,however, being able to closely control the shape of the micro-scatterersor the scattering pattern. The simplest screens have been in the form ofclassical ground glass diffusers. Other devices have includedholographic diffusers and microlens arrays. In most of these cases, someelement of randomness has been introduced by the recording of a specklepattern or by superposing and distributing microlens shapes in a randomfashion.

[0024] Thus, as with diffusing screens, there exists a need in the artfor well-controlled display screens with elementary microstructures thatare well defined and chosen to meet specific diffusion requirements.

III. SUMMARY OF THE INVENTION

[0025] In view of the foregoing, it is an object of the presentinvention to provide a structured screen surface that addresses thedrawbacks mentioned above by allowing control of the elementarystructures (microstructures) that define the surface as well as theirrelative distribution across the surface of the device. The control ofsurface shape and relative spatial arrangement is completelydeterministic, in contrast with the prior art which relies onstatistical control and placement of microstructures. With the approachof the invention it is possible to modify the quality of the lightspreading process so as to make it appropriate and useful in a widerange of applications.

[0026] A key distinction of the invention in relation to the prior artis its ability to finely shape the form of the screen surface andarbitrarily distribute it to within, for example, fractions of a micronas well as position the elementary units relative to each other in aprecise and general fashion. Methods based on statistical processes areunable to attain such accuracy and, for that reason, can control thespreading pattern only to a limited degree. Put simply, the precisedefinition of the light spreading (scattering) pattern produced by ascreen depends on the features of the screen. If only limited control ispossible over the screen structure, only limited shaping is possibleover the resulting scattering.

[0027] The structured relief surface according to the invention iscomposed of two main aspects, which are the shape of the elementaryunits and the spatial distribution of such units. The particular shapeof the elementary units is defined by the required properties of thelight spreading. The specific shapes can assume many forms including,but not limited to, spherical, parabolic, hyperbolic, piecewise linear,piecewise polynomial, pyramidal, conical or combinations thereof.Specific shapes affect the spreading differently and specific choicescan be made that are suitable to the spreading required for anyparticular application. Combinations of different shapes at differentlocations on a screen surface can be used if desired.

[0028] In addition to the shape of the elementary units, the relativespatial arrangement of those units can be defined arbitrarily. For thispurpose, we introduce two distinct coordinate systems that completelydefine the surface relief of the screen. In relation to these frames ofreference, it is possible to define an arbitrary spatial distribution ofmicrostructures relative to a global frame but also relative to itslocal reference coordinate system. In the present inventive process, thearrangement of units follows any specified pattern with a precision of,for example, only a fraction of a micron. A key distinction of thepresent invention in relation to the prior art is its ability todistribute elementary units according to patterns where not only shapesand sizes but also relative locations are distinct.

[0029] In accordance one of its aspects, the invention provides a methodfor making a structured screen that provides a desired spread ofincident light, said structured screen comprising a substrate and aplurality of microstructures distributed over at least one surface ofsaid substrate, said method comprising:

[0030] (a) selecting a location on said at least one surface of thesubstrate for each of said plurality of microstructures;

[0031] (b) selecting a configuration for each of said plurality ofmicrostructures;

[0032] (c) calculating the spread of the incident light for the selectedlocations and the selected configurations of steps (a) and (b);

[0033] (d) comparing the calculated spread of step (c) with the desiredspread and, if necessary, repeating at least one of steps (a) and (b),and step (c) until the comparison between the calculated spread anddesired spread satisfies a specified criterion (e.g., angular spread,homogeneity, etc.); and

[0034] (e) producing a plurality of microstructures having, to anaccuracy of better than 10·λ_(n), the locations and the configurationswhich, in step (d), resulted in the satisfaction of the specifiedcriterion, where 10·λ_(n) is the nominal operating wavelength for thescreen.

[0035] In accordance with another of its aspects, the invention providesapparatus for controlled spreading of light comprising a plurality ofmicrostructures, each microstructure being located with better than10·λ_(n) accuracy at a predetermined location with respect to all othermicrostructures and each microstructure having a configuration thatcorresponds, with better than 10·λ_(n) accuracy, to a predeterminedmathematical relation, where λ_(n) is the nominal operating wavelengthof the apparatus and said predetermined locations and predeterminedmathematical relations allow an a priori calculation of the spreading ofincident light by the apparatus.

[0036] The nominal operating wavelength λ_(n) can be, for example, themidpoint of the wavelength range over which the screen (apparatus) willbe used or a particular wavelength of interest within such a range.Preferably, the accuracy of the microstructures and their locations isbetter than 5·λ_(n), and most preferably better than λ_(n) or fractionsthereof. For a screen that is to be used in the visible range, thiscorresponds to a preferred accuracy on the order of a few microns and amost preferred accuracy on a sub-micron (fraction of a micron) level. Tosimplify the presentation, the most preferred level of accuracy for thevisible range is referred to at various points in the specification, itbeing understood that these references are only for the purpose offacilitating the discussion of the invention and are not intended in anyway to limit the invention to this level of accuracy.

[0037] In certain embodiments, at least a portion of at least some ofthe microstructures is selected to have a configuration given by:${s\left( {x,y} \right)} = {\frac{c\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}{1 + \sqrt{1 - {\left( {\kappa + 1} \right){c^{2}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}}}} + {\sum\limits_{p}{A_{p}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}^{p/2}}}$

[0038] where s(x,y) is the sag of said portion, c is its curvature,(x_(c), y_(c)) is its center point, κ is a conic constant, and A_(p) areaspheric coefficients. In certain aspects of these embodiments, A_(p)≠0for at least one p, or κ≠0, or κ=−1 and A_(p)=0 for all p.Microstructures having at least one of these three properties (i.e.,A_(p)≠0 for at least one p, or κ≠0, or κ=−1 and A_(p)=0 for all p) arethemselves an aspect of the invention.

[0039] In other embodiments, at least a portion of at least some of themicrostructures is selected to have a configuration given by:${s\left( {x,y} \right)} = {{\sum\limits_{p = 1}^{\infty}{B_{p}\left( {x - x_{c}} \right)}^{p}} + {C_{p}\left( {y - y_{c}} \right)}^{p}}$

[0040] where s(x,y) is the sag of said portion, (x_(c), y_(c)) is itscenter point, and B_(p) and C_(p) are power series coefficients.

[0041] In further embodiments, at least some of the microstructurescomprise an anamorphic microlens. In connection with these embodiments,at least a portion of at least some of the microstructures can beselected to have a configuration given by:${s\left( {x,y} \right)} = \frac{{c_{x}\left( {x - x_{c}} \right)}^{2} + {c_{y}\left( {y - y_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{x}} \right){c_{x}\left( {x - x_{c}} \right)}^{2}} + {\left( {1 + \kappa_{y}} \right){c_{y}\left( {y - y_{c}} \right)}^{2}}}}$

[0042] where s(x,y) is the sag of said portion, (x_(c), y_(c)) is itscenter point, c_(x) and c_(y) are curvatures along x and y,respectively, and κ_(x) and κ_(y) are conic constants along x and y,respectively.

[0043] Alternatively, for these embodiments, the configuration can begiven by:${s\left( {x,y} \right)} = {\frac{{c_{x}\left( {x - x_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{x}} \right)\left( {x - x_{c}} \right)^{2}}}} + \frac{{c_{y}\left( {y - y_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{y}} \right)\left( {y - y_{c}} \right)^{2}}}} + {\sum\limits_{p}{A_{xp}\left( {x - x_{c}} \right)}^{p}} + {A_{yp}\left( {y - y_{c}} \right)}^{p}}$

[0044] where s(x,y) is the sag of said portion, (x_(c), y_(c)) is itscenter point, c_(x) and c_(y) are curvatures along x and y,respectively, κ_(x) and κ_(y) are conic constants along x and y,respectively, and A_(xp) and A_(yp) are higher order asphericcoefficients along x and y, respectively.

[0045] The first form is generally not used with aspheric coefficientssince it couples curvatures and conic constants in x and y.

[0046] In still further embodiments, at least some of themicrostructures comprise a curved, microlens portion and astraight-sided, piston (cylindrical) portion. Microstructures havingthis microlens-piston structure are themselves an aspect of theinvention.

[0047]FIG. 1 schematically illustrates two microstructures constructedin this way, where microstructure 1 has a spherical microlens portion ofmaximum sag s₁ and a piston portion of diameter d₁ and height (offset)p₁, and microstructure 2 has a parabolic microlens portion of maximumsag s₂ and a piston portion of diameter d₂ and height (offset) p₂, withthe apices of the two microstructures being separated by a depth D. Forease of illustration, this drawing shows convex microstructures and onlytwo microlens configurations, it being understood that the inventionalso applies to concave microstructures, combinations of convex andconcave microstructures, and microlens configurations of any and alltypes.

[0048] In other embodiments, at least a portion of at least some of themicrostructures is selected to have a configuration characterized by atleast one parameter with the at least one parameter being randomlydistributed in accordance with a predetermined probability densityfunction (e.g., a uniform probability density function over apredetermined range for the parameter). Screens having microstructureshaving such randomized configurations are themselves an aspect of theinvention.

[0049] Examples of parameters which can be randomly distributed include:

[0050] radius of curvature; maximum surface sag; a parametercharacteristic of the transverse size of a microstructure (e.g.,diameter); for microstructures which comprise a curved, microlensportion and a straight-sided, piston portion (see FIG. 1), the heightsof the straight-sided, piston portions; and for microstructures havingapices, the distances of the apices from the screen's substrate or,alternatively, where the distances have a maximum value, the differencesbetween the distances and said maximum value.

[0051] More than one parameter (e.g., two parameters) can be randomized,if desired, with the randomization (e.g., probability density function)being the same or different for the parameters. For example, in the caseof a microstructure which comprises a curved, microlens portion and astraight-sided, piston portion, one randomly-distributed parameter cancharacterize the curved, microlens portion and a secondrandomly-distributed parameter can characterize the straight-sided,piston portion.

[0052] In additional embodiments, the locations of the microstructuresform a regular array (e.g., a hexagonal array). In other embodiments,the locations are based on a set of unit cells which form a mosaic(e.g., a random mosaic). In connection with these embodiments, thescreen can have internal microstructures and edge microstructures, withthe mosaic providing at least some junctions between internalmicrostructures that correspond, in terms of light spreading, to atleast some junctions between edge microstructures resulting from thetiling of two screens to one another.

[0053] In further embodiments, the locations of the microstructures arerandomly distributed in accordance with a predetermined probabilitydensity function. For example, the locations of the microstructures canbe based on a random set of polygonal shaped boundaries.

[0054] In accordance with still further embodiments, the substrate ofthe screen comprises two spaced-apart (e.g., major) surfaces and themicrostructures are distributed over both surfaces. In otherembodiments, microstructures are distributed over one of the surfaces,with the other surface comprising a Fresnel lens.

[0055] In additional embodiments, the screen's substrate defines a firstoptical axis and the configuration of at least some of themicrostructures comprises a microlens which defines a second opticalaxis which is not parallel to the first optical axis. Screens havingsuch a configuration are themselves an aspect of the invention.

[0056] In accordance with another of its aspects, the invention providesa structured screen comprising a plurality of predeterminedmicrostructures, wherein:

[0057] (a) said microstructures comprise a curved, microlens portion anda straight-sided, piston portion which has a predetermined height whichcan be zero:

[0058] (b) said curved, microlens portions have predetermined diametersand predetermined maximum sags; and

[0059] (c) for at least some of said microlenses, the sum of thepredetermined maximum sag and the predetermined height is greater thanthe predetermined diameter.

[0060] In accordance with this aspect of the invention, at least one ofthe predetermined diameters, the predetermined maximum sags, and thepredetermined heights can be randomly distributed in accordance with apredetermined probability density function (e.g., a uniform probabilitydensity function over a predetermined range for said diameters, maximumsags, and/or heights).

[0061] In accordance with a further of its aspects, the inventionprovides a structured screen comprising a plurality of predeterminedaspherical microlenses (e.g. parabolic microlenses), wherein saidmicrolenses:

[0062] (a) have predetermined diameters and predetermined maximum sags;and

[0063] (b) produce a spread of incident light which has a flatterintensity distribution than that produced by a plurality of sphericalmicrolenses having the same predetermined diameters and predeterminedsags.

[0064] In accordance with this aspect of the invention, at least one ofthe predetermined diameters and the predetermined maximum sags can berandomly distributed in accordance with a predetermined probabilitydensity function (e.g., a uniform probability density function over apredetermined range for said diameters and/or maximum sags).

[0065] In accordance with another of its aspects, the invention providesa structured screen comprising:

[0066] (a) a Fresnel lens which comprises a plurality of surfaces in theform of concentric rings; and

[0067] (b) a plurality of microstructures distributed over at least someof said plurality of surfaces, said plurality of microstructures servingto control the spread of light incident on the structured screen.

[0068] In accordance with a further aspect, the invention provides astructured screen comprising a plurality of unit cells and a pluralityof microstructures, one microstructure associated with each unit cell,wherein the perimeters of the unit cells are non-regular polygons. Incertain embodiments of this aspect of the invention, the perimeters canbe defined by a predetermined probability density function.

[0069] In accordance with an additional aspect, the invention provides astructured screen comprising a plurality of microstructures at leastsome of which comprise a microlens having a first curvature in a firstdirection and a second curvature in a second direction orthogonal to thefirst direction, at least one of said first and second curvatures beingrandomly distributed in accordance with a predetermined probabilitydensity function. In certain embodiments of this aspect of theinvention, both the first and second curvatures are randomly distributedin accordance with a predetermined probability density function whichmay be the same or different for the two curvatures.

[0070] In accordance with a further aspect, the invention provides astructured screen comprising:

[0071] (a) a first sub-screen comprising a plurality of internalmicrostructures and a plurality of edge microstructures, eachmicrostructure being located at a predetermined location with respect toall other microstructures, said predetermined locations being based on afirst set of unit cells which form a first mosaic; and

[0072] (b) a second sub-screen comprising a plurality of internalmicrostructures and a plurality of edge microstructures, eachmicrostructure being located at a predetermined location with respect toall other microstructures, said predetermined locations being based on asecond set of unit cells which form a second mosaic;

[0073] wherein:

[0074] (i) the first and second sub-screens are tiled to one another,said tiling producing edge junctions between edge microstructures of thefirst sub-screen and edge microstructures of the second sub-screen; and

[0075] (ii) each of the first and second mosaics provides at least someinternal junctions between internal microstructures that correspond, interms of light spreading, to at least some of the edge junctions.

[0076] In certain embodiments of this aspect of the invention, each ofthe first and second mosaics can be random. In other embodiments, thefirst and second sub-screens are identical.

[0077] The advantages of the various aspects and embodiments of theinvention referred to above will become apparent in the drawings anddetailed description of the invention which follow.

IV. BRIEF DESCRIPTION OF THE DRAWINGS

[0078]FIG. 1 is a schematic drawing illustrating representativemicrostructures of the invention.

[0079]FIG. 2 shows diffraction patterns due to two periodic arrays ofmicrolenses of depth equal to 24 μm, calculated using Eq. (2). Thenominal operating wavelength is λ_(n)=0.6328 μm and the period is Λ=200μm. The dashed line is due to a spherical microlens with curvature equalto 0.0157 μm⁻¹ while the solid line employs the following parameters(see Eq. (4)): c=0.0118 μm⁻¹ and κ=−1.09, without the inclusion ofaspheric coefficients.

[0080]FIG. 3 is a comparison of the elementary units of the arrays usedto calculate the diffraction patterns shown in FIG. 1. The solid linerefers to the aspherical surface with nonzero conic while the dashedline refers to the spherical surface.

[0081]FIG. 4 shows the difference between the spherical and asphericalprofile necessary to generate the diffraction pattern shown in FIG. 1.

[0082]FIG. 5 is a calculated diffusion pattern for a regular array ofparabolic units.

[0083]FIG. 6 is a calculated diffusion pattern for a random array ofparabolic units with variable depth and constant pitch.

[0084]FIG. 7 is a calculated diffusion pattern for a random array ofparabolic units with variable depth and variable pitch.

[0085]FIG. 8 is a histogram of radii of curvature for a singlerealization of a random microlens array having a uniform probabilitydensity function for radius of curvature.

[0086]FIG. 9 is a histogram of depth for a single realization of arandom microlens array having a uniform probability density function forradius of curvature.

[0087]FIG. 10 is a calculated three-dimensional scattering pattern dueto a random microlens array with a hexagonal unit cell structure. Theprobability density function is uniform for radius of curvature.

[0088]FIG. 11 is a histogram of radii of curvature for a singlerealization of a random microlens array with a uniform probabilitydensity function for microlens depth.

[0089]FIG. 12 is a histogram of depth for a single realization of arandom microlens array with a uniform probability density function formicrolens depth.

[0090]FIG. 13 is a calculated three-dimensional scattering pattern dueto a random microlens array with a hexagonal unit cell structure. Theprobability density function is uniform for microlens depth.

[0091]FIG. 14 is a schematic diagram of coordinate systems that can beused to define the spatial localization of elementary structures(microstructures) on one or more surfaces of a screen. For simplicityonly a two-dimensional plot is shown. The (X,Y) frame denotes a globalcoordinate system, while the (x,y) frames denote local coordinatesassociated with individual microstructures.

[0092]FIG. 15 illustrates elementary units arranged on a regular squarelattice with each unit (represented by circles) centered on a unit cell(square regions containing circles) and having a size that is no greaterthan the cell itself. The center of a cell is represented by the crossmark.

[0093]FIG. 16 illustrates an arrangement of elementary units on a squarelattice of constant fill factor but with centers of individual unitsdisplaced with respect to the centers of the unit cells.

[0094]FIG. 17 illustrates an arrangement of elementary units on a squarelattice of variable fill factor and with centers of individual unitsdisplaced with respect to the centers of the unit cells.

[0095]FIG. 18 illustrates an arrangement of elementary units on a squarelattice with 100% fill factor. The boundary of each elementary unit isdenoted by the dotted line but the unit itself is only defined withinthe square unit cell. The center of individual units may be displacedwith respect to the center of the cell.

[0096]FIG. 19 illustrates elementary units arranged on a regularhexagonal lattice with 100% fill factor.

[0097]FIG. 20 illustrates elementary units distributed over a random setof polygonal shaped unit boundaries. In this figure, only a view of thecontours of the boundaries is shown, in the form of a Voronoi diagram(see Lectures on Random Voronoi Tessellations, Jesper Møller, New York:Springer-Verlag, 1994).

[0098]FIG. 21 illustrates a structured screen where both surfaces of asubstrate of thickness τ are structured with an array of individualelementary units, according to some of the embodiments of the presentinvention.

[0099]FIG. 22 shows calculated diffraction patterns for diffusive screenarrays composed of triangular elementary units with a vertical offset(solid line) compared to a regular array (dashed line) without such anoffset.

[0100]FIG. 23 is a schematic drawing showing an illumination source, aFresnel lens, and microlens array.

[0101]FIG. 24 is a schematic drawing showing a prior art integration ofa screen and a Fresnel lens.

[0102]FIG. 25 is a schematic drawing showing the combination ofdiffusion and Fresnel collimation on a single surface.

[0103]FIG. 26 shows a calculated scattering pattern for a regular arrayof spherical microlenses 10-μm deep for operation in the infrared.

[0104]FIG. 27 shows a calculated scattering pattern for a regular arrayof triangular microstructures 10-μm deep for operation in the infrared.

[0105]FIG. 28 shows a calculated scattering pattern for a regular arrayof hyperbolic microstructures 10-μm deep for operation in the infrared.

[0106]FIG. 29 shows a calculated scattering pattern for a regular arrayof spherical microstructures 12-μm deep for operation in the visible.

[0107]FIG. 30 shows a calculated scattering pattern for a regular arrayof spherical microstructures 20-μm deep for operation in the visible.

[0108]FIG. 31 shows a calculated scattering pattern for a regular arrayof parabolic microstructures 20-μm deep for operation in the visible.

[0109]FIG. 32 shows a regular array of spherical microlenses (diameter:100 microns, maximum sag: 10 microns).

[0110]FIG. 33 shows an array of identical spherical microlenses(diameter: 100 microns, maximum sag: 10 microns), with random verticaloffset equal to ±2 microns.

[0111]FIG. 34 shows a calculated scattering profile for an array ofidentical spherical microlenses of 100 microns diameter and 10 micronsmaximum sag. Vertical microlens positioning includes a maximum verticaloffset of +2 microns.

[0112]FIG. 35 shows calculated scattering patterns for arrays ofidentical spherical microlenses of diameter 100 microns and maximum sag10 microns for operation in the visible, with and without a verticaloffset of ±2 microns.

[0113]FIG. 36 illustrates a square arrangement of microstructures.

[0114]FIG. 37 illustrates a hexagonal arrangement of microstructures.

[0115]FIG. 38 illustrates a random arrangement of microstructuresdelimited by random polygonal boundaries.

[0116]FIG. 39 illustrates a screen with a mosaic spatial arrangement.Note the relative displacement of microlenses and the tiling naturallyexhibited by this configuration, i.e., the mosaic has at least somejunctions between internal microstructures that correspond, in terms oflight spreading, to at least some junctions between edge microstructureswhich would result from the tiling of two screens to one another.

[0117]FIG. 40 shows frequency of sag and frequency of radius ofcurvature (PDF uniform in radius of curvature) for a sag range of 5-15μm and 100-micron diameter spherical microlenses.

[0118]FIG. 41 shows frequency of sag and frequency of radius ofcurvature (PDF uniform in sag) for a sag range of 5-15 μm and 100-microndiameter spherical microlenses.

[0119]FIG. 42 shows calculated scattering patterns which illustrate theeffect of a probability distribution function on scattering pattern foran array of random spherical microlenses with a sag range between 5 μmand 15 μm. The lens diameter is 100 microns and the vertical offset is±2 μm.

[0120]FIG. 43 shows an array of spherical microlenses with a hexagonalarrangement. Each microlens is 0.5 mm in diameter and has a maximum sagin the range 2 to 8 μm. There is also a vertical offset of ±2 μm.

[0121]FIG. 44 shows a calculated three-dimensional scattering patternobtained with the structured screen shown in FIG. 43.

[0122]FIG. 45 shows cross-sections of the scattering pattern of FIG. 44along two perpendicular directions.

[0123]FIG. 46 is a calculated scattering pattern for a hexagonal displayscreen.

[0124]FIG. 47 shows a hexagonal array of anamorphic microlenses (averagediameter 50 μm). The bar indicates microlens depth.

[0125]FIG. 48 shows a calculated scattering pattern for an anamorphicmosaic array.

[0126]FIG. 49 shows a mosaic array of anamorphic microlenses. The barindicates microlens depth.

V. DETAILED DESCRIPTION OF THE INVENTION

[0127] A. Diffraction Equation The structured screens of the inventionare designed using diffraction equations appropriate to the conditionsunder which the finished screen is to be used (near field, far field,with or without focusing lenses, etc.).

[0128] For illustration purposes consider a structured screen which isto operate in the far field under illumination of wavelength λ. Thefield fat a point (u,v) at the observation plane is given by the Fouriertransform of the surface-relief structure or shape of the structuredscreen s(x,y), given by (Introduction to Fourier Optics, J. W. Goodman,McGraw-Hill Publishing Company, New York, 1968) $\begin{matrix}{{{f\left( {u,v} \right)} = {\frac{\exp ({ikz})}{i\quad \lambda \quad z}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{\exp \left\lbrack {{{{ik}\left\lbrack {{n(\lambda)} - 1} \right\rbrack}{s\left( {x,y} \right)}} - {i\quad \frac{2\pi}{\lambda \quad z}\left( {{xu} + {yv}} \right)}} \right\rbrack}{x}{y}}}}}},} & (1)\end{matrix}$

[0129] where k=2π/λ defines the magnitude of the wavevector and n(λ)gives the index of refraction at wavelength λ. The coordinates (x,y)define a point in the plane of the screen.

[0130] The integral over the whole surface can be broken into integralsover N cells D_(j), j=1, . . . , N, that completely cover the substratesurface. In its most general form, the shape of a microstructurecontained in each cell is expressed as a local surface shape s_(j), j=1,. . . , N, where in general s_(p)≠s_(q), for p≠q. The field f can now bewritten as $\begin{matrix}{{f\left( {u,v} \right)} = {\frac{\exp ({ikz})}{i\quad \lambda \quad z}{\sum\limits_{J}{\int_{D_{j}}{\int{{\exp \left\lbrack {{{{ik}\left\lbrack {{n(\lambda)} - 1} \right\rbrack}{s_{j}\left( {x,y} \right)}} - {i\quad \frac{2\pi}{\lambda \quad z}\left( {{xu} + {yv}} \right)}} \right\rbrack}{x}{{y}.}}}}}}} & (2)\end{matrix}$

[0131] The above expression illustrates the significant elementsinvolved in the design of the structured screens of the invention. Thefirst element is the shape of individual elements s_(j)(x,y). The secondis the cell D_(j), where the individual element is contained. The priorart has had limited control over the elementary functions s_(j)(x,y)except to guarantee some amount of focusing power, and the cells haveusually been square or hexagonal.

[0132] The present invention improves upon previous approaches by (i)allowing exact definition of the shape s_(j)(x,y) to, for example, asub-micron accuracy; (ii) allowing precise definition of the spatiallocalization of the domain of each cell D_(j) relative to any othercell; and (iii) allowing precise definition of the cell shape D_(j), notlimited to only square or hexagonal arrays, but assuming any contournecessary to implement the desired spreading pattern.

[0133] B. Surface Definition

[0134] (1) Microstructures Per Se

[0135] The surface-relief pattern on any one side or on both sides ofthe structured screen is responsible for the spreading of the incidentillumination. This pattern achieves the goal of shaping the diffusedillumination by a specified arrangement and choice of elementary units,which are generally different from each other and are displaced atspecific distances from each other, according to the requirements of thedesired beam shaping.

[0136] In a general form, the shape of each elementary unit can bedefined in terms of segments characterized by piecewise functions. Thegeneral shape can assume either a continuous or discontinuous form andcan also impart focusing power to the incident illumination. Theparticular shape of each individual element is of fundamental importancein shaping the diffused light. Thus, in accordance with the invention,the elementary shape is precisely control to tailor the diffusionprofile.

[0137] As an illustration, FIG. 2 shows diffusion patterns for of tworegular arrays of identical relief depth but with the shape of eachelementary unit in one array being different from the shape of each unitin the other array. In both cases the total relief depth is the same.The difference in the patterns is evident in this figure.

[0138] A comparison between the profiles of each individual shape forthe two arrays is shown in FIG. 3, while the difference between the twoprofiles is shown in FIG. 4. Note that to achieve the kind of diffusioncontrol illustrated by the two curves of FIG. 2, it is necessary toensure microstructure relief control on the order of, for example, a fewmicrons.

[0139] To further emphasize the importance of the shape of theelementary units, we note from grating theory that, while the gratingperiod and wavelength determines the divergence angles of diffractedorders, the grating profile determines the distribution of power amongorders. For example, a triangular blazed grating can be designed for100% theoretical efficiency at a single order. On the other hand, customshapes can be obtained that spread the incident illumination over manyorders. Such control of the light spread is only possible with properchoice of the elementary shape of the grating.

[0140] As discussed above, the prior art commonly relies on elementaryunits of spherical shape to accomplish light diffusion. The sphericalshape can be expressed according to the following equation$\begin{matrix}{{{s\left( {x,y} \right)} = {\frac{1}{c} - \sqrt{\frac{1}{c^{2}} - \left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}}},} & (3)\end{matrix}$

[0141] where c denotes the curvature of the surface and (xc,yc) is thecenter point. As can be seen in this equation, the only degree offreedom available to control the diffusion is the curvature or thefocusing power.

[0142] To introduce additional degrees of freedom and to allow for awider class of surfaces, the surface shape can be written as follows:$\begin{matrix}{{{s\left( {x,y} \right)} = {\frac{c\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}{1 + \sqrt{1 - {\left( {\kappa + 1} \right){c^{2}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}}}} + {\sum\limits_{p}{A_{p}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}^{p/2}}}},} & (4)\end{matrix}$

[0143] where one finds the conic constant κ and the aspheric terms{A_(p)} as new degrees of freedom, as compared to Eq. (3). The sphericalshape is obtained in the particular case of κ=0 and A_(p)=0, for all p.With more degrees of freedom, it becomes possible to control thediffusion pattern better so as to satisfy specific system requirements.

[0144] Although Eq. (4) allows a wide variety of shapes to beimplemented, it is limited to conic surfaces with aspheric corrections.However, in accordance with the invention, any surface that can bedefined by a mathematical relation can be implemented, includingmathematical relations based on algorithmic processes. In general, thesurface can be specified as a piecewise function over the boundary ofthe microstructure such that within each interval the surface can beexpanded in a power series of the form: $\begin{matrix}{{{s\left( {x,y} \right)} = {{\sum\limits_{p = 1}^{\infty}{B_{p}\left( {x - x_{c}} \right)}^{p}} + {C_{p}\left( {y - y_{c}} \right)}^{p}}},} & (5)\end{matrix}$

[0145] where (x_(c),y_(c)) is the center point and s(x,y) is definedover a limited area of the microstructure. The total function over thecomplete area of the microstructure would then be defined in a piecewisemanner.

[0146] In addition to the foregoing, the present invention also allowsthe profile of each micro structure to vary across all or part of anarray with each microstructure still being controlled to, for example,sub-micron accuracy. In this way it is possible to homogenize thediffused light and avoid the visual artifacts caused by a periodicarray.

[0147] The implementation of regular arrays has some advantages from afabrication point of view but the performance of the screen may not besatisfactory due to the introduction of image artifacts such as high-frequency intensity variations. Furthermore, the control of lightdiffusion is limited because an array in general will not meet all typesof system criteria. In some broadband applications, regular arrays maybe acceptable because the smoothing by spectral dispersion helps tominimize the image artifacts due to the grating structure.

[0148] The behavior of a regular array is shown in FIG. 5 for parabolicmicrolenses of diameter equal to 100 μm and depth 5 μm. The incidentillumination belongs to the 400-700 nm spectral band. One observes thatalthough diffusion is accomplished, high-frequency intensity variationsare present and higher-order components are noticeable.

[0149]FIG. 6 shows the diffusion obtained by a regular parabolic arrayof diameter 100 μm but with depth randomly chosen in the range 5-10 μm,averaged over several statistically identical screen configurations. Theintensity diffusion can be well-described by a fourth-ordersupergaussian with an angular spread of about 7.7 degrees measured atthe 1/e² intensity point. The randomization avoids the high-frequencystructure as well as high order diffraction angles. Note that thesecalculations assume a spatially coherent beam incident on the aperture.In practice, partial coherence effects would further reduce theintensity fluctuations.

[0150]FIG. 7 shows another diffusion pattern of parabolic units withrandom depth in the range 5-10 μm and with the diameter sizes of eachunit randomly chosen with a variation in the range ±20% with respect tothe nominal diameter of 100 μm. The main difference with respect to thepattern of FIG. 6 is the elongated tail of Lorentzian shape.

[0151] Control of the diffusion process depends on the particular waymicrostructures differ from each other. In general, this variability canbe expressed in terms of a probability distribution function (PDF) for aparticular parameter(s) of choice. For instance, if the array presentsmicrolens structures with random radii of curvature, then there exists aPDF that defines how the radius of curvature varies across the array.Similarly, the PDF might refer to a conic constant, the depth of a givenmicrostructure, the location of a given microstructure, the size of agiven microstructure, any combination of these parameters, or any otherrelevant parameters or combinations thereof.

[0152] In all cases, the PDF can be arbitrarily defined and an array canbe accordingly built. The exact correspondence between the actual arrayand the PDF that describes it requires that each elementarymicrostructure be fabricated with, for example, sub-micron accuracy.Methods for specifying a PDF depend largely on the desired properties ofthe diffusion pattern but can be of either a deterministic orstatistical nature.

[0153] To illustrate the effect of the particular choice of a PDFconsider a two-dimensional array of spherical microlenses with ahexagonal arrangement. Each microlens has an external circumferencewhose diameter is 750 μm and a total sag in the range 4-16 μm. Firstconsider the case where the PDF is uniform in radius of curvature. For agiven realization of the array, the histograms of radii and microlenssag are shown in FIGS. 8 and 9, respectively. The correspondingdiffusion pattern is shown in FIG. 10.

[0154] Now consider a random array with the same sag range but with auniform PDF in sag instead of radius of curvature. For a givenrealization of the array, the histograms of radii and sag are shown inFIGS. 11 and 12, respectively. The corresponding diffusion pattern isshown in FIG. 13. The difference in the PDF for these two cases leads tofundamentally distinct arrays and the difference is reflected in theresulting diffusion, as can be readily seen by comparing FIG. 10 withFIG. 13. (2) Distribution of Microstructures

[0155] As described above, an accurate description of the screen surfaceor surfaces requires two basic elements. The first element is themicrostructure itself, which can be of any particular shape, asdiscussed immediately above. The second element is the relative spatialplacement of the individual microstructures where each microstructure isspatially placed with respect to each other arbitrarily and with, forexample, sub- micron accuracy. These two elements, together with theproperties of the light that illuminates the screen, determine in aunique way the diffusive features of the screen. While the shape of eachmicrostructure has a predominant effect over the functional propertiesof the diffusion, the spatial placement of microstructures determinesthe spatial symmetry, or lack thereof, of the diffusion.

[0156] To define precisely the spatial placement of the microstructureson the surfaces of the screen, we define two basic sets of coordinatesystems in relation to the screen. The global coordinate system can belocated arbitrarily with respect to the screen and defines a referenceframe from which each microstructure can be localized with respect tosome arbitrary reference point such as the vertex, tip, or any otherelement of the microstructure.

[0157] A local coordinate system is next associated with eachmicrostructure. In the local reference frame the surface shape of themicrostructure can be defined according to a function of the forms(x−X_(k),y−Y_(k)), where s denotes the functional form of themicrostructure, (x,y) denotes a point in the local coordinate system,and (X_(k),Y_(k)) is the position of the k^(th) local coordinate systemwith respect to the global reference frame where k runs from 1 to thetotal number of microstructures present on the screen. A schematicillustration of the two reference frames is shown in FIG. 14, where forsimplicity, the two dimensional case is shown, it being understood thatthe general case is three dimensional.

[0158] In relation to the global coordinate system one may distinguishmicrostructure positioning along the surface of the screen orperpendicular to it. Along the surface, various realizations of thisembodiment include regular periodic arrays, random arrays with thedistance between neighbor microstructures varying as a function ofposition, microstructures with well-defined boundaries such as in asquare or hexagonal array, microstructures with random boundaries wherethe size and shape of each microstructure varies as a function ofposition across the screen, or structures positioned on both surfaces ofthe screen.

[0159] Examples of possible arrangements are illustrated in FIGS. 15through 21. For cases which include a random component, such componentwill be defined in terms of a probability density function (PDF).

[0160] In the direction perpendicular to the screen surface, there canbe a vertical offset of each microstructure with respect to each other,also called piston. The presence of piston is relevant in the reductionof high-frequency intensity variations in the diffusion pattern(speckle), as well as in avoiding the presence of hot spots, which areisolated regions of the diffusion pattern that exhibit much higherintensity than the average of the whole pattern.

[0161] The amount of piston used in any particular application willdepend on the characteristics of the illumination on the screen but, ingeneral, should be equivalent to a few optical wavelengths of theillumination. Also, the magnitude of the piston component for neighbormicrostructures will typically vary randomly according to a PDF, unlessit is desirable to introduce some bias in the diffusion pattern.

[0162]FIG. 22 illustrates the effect of a vertical offset on adiffraction pattern for an elementary unit having a triangular shape.For a regular array having a grating period of 200 μm and a depth of 5μm, light of wavelength 0.5 μm is focused at a diffracted order centeredat the angular distance of −0.14 degrees with an angular divergence of0.006 degrees. The corresponding diffraction pattern is shown in FIG. 22by the dashed curve. The solid curve, on the other hand, is the resultof an ensemble average over statistically identical arrays of triangularunits with a maximum offset (piston) of 2 μm. The offset obeys a uniformprobability distribution function.

[0163] The diffusion attained through the introduction of the offset ismore that one order of magnitude. Note however that this level ofdiffusion is peculiar to the blazed grating that was analyzed. Ingeneral, the degree of diffusion is shape-dependent. However, as ageneral rule, the offset helps in the smoothing of the diffusionpattern, since it helps eliminate artifacts due to the periodic gratingstructure.

[0164] (3) Fresnel Function

[0165] In addition to their diffusion function, the distributedmicrostructures of the invention can be used in combination with or,indeed, can constitute a Fresnel lens, whose purpose is to collimate anotherwise divergent beam of light.

[0166] In the prior art, the use of a Fresnel lens is generally assumedand the general setup is as shown in FIG. 23. There has been someattempts to incorporate the Fresnel capability on one side of asubstrate with the diffusive feature associated with the second surfaceas illustrated in FIG. 24. The screens of the present invention can beused with Fresnel lenses of the type shown in either of these figures.

[0167] However, these approaches require several processing steps toprovide the final screen with both capabilities at different surfaces.According to the present invention, a reduction in such processing stepscan be achieved by incorporating both a diffusing function and a Fresnelfunction on one screen surface.

[0168] As discussed above, in accordance with the invention, one isallowed precise control of each microstructure and its spatiallocalization in the frames of reference that define the structuredscreen. In addition to translations and piston as components of thespatial placement, one can also rotate individual elements.Significantly, such rotation allows one to achieve the function of aFresnel lens.

[0169] In particular, as illustrated in FIG. 25, the function of aFresnel lens in a structured screen simply requires rotation ofindividual microstructures having focusing power. As can be seen in thisfigure, the individual microlenses have optical axes which are notparallel to the optical axis of the overall screen. Although notexplicitly shown in this figure, the optical axis of the screen istypically the optical axis of the screen's substrate. By orienting themicrostructures in this way, the diffusive and Fresnel-collimationfeatures can not only be integrated on a single substrate but on asingle surface, thereby reducing the number of processing steps requiredto generate the screen and allowing large volume replication in a singlestep.

[0170] C. Fabrication

[0171] The ability to produce highly reproducible and accurate screensurfaces of the type described above requires a fabrication method thatallows screens to be manufactured consistently and with point-by-pointaccuracy. Although other techniques can be used, the most suitablemethod for such a task is direct laser writing where a laser beam scansa properly prepared substrate with variable intensity.

[0172] In such a method, a substrate such as glass is covered with, forexample, a low-contrast photosensitive polymer (photoresist) that isexposed and records in a latent image the exposure pattern defined bythe laser beam. Typically, the photosensitive material is positive, inwhich case, when the substrate is developed, the exposed material iseliminated leaving a surface relief structure.

[0173] This surface relief structure basically defines the desiredscreen surface or, in some cases, its complement. See commonly assignedU.S. patent application Ser. No. 60/222,032 which is being filedconcurrently herewith in the names of Geoffrey B. Gretton, G. MichaelMorris, and Tasso R. M. Sales, and is entitled “Microlens Arrays HavingHigh Focusing Efficiency,” the contents of which in its entirety isincorporated herein by reference.

[0174] The surface relief structure obtained upon development of thephotoresist may not be precisely the desired structure depending onspecific performance characteristics of the fabrication process. Thatis, the fabrication process itself can introduce features that may beundesirable in the final product and need to be accounted for in usingthe process. For example, since a writing laser beam has a finite size,the final surface after development represents the profile defined bythe laser exposure system convolved with the shape of the laser beamused. In some instances the presence of convolution may not bedetrimental to the performance of the screen but in other cases it mustbe avoided. Of course, the performance requirements and operatingconditions dictate the fabrication tolerances and limitations.

[0175] The surface relief profile left on the surface of the substratecan serve as a master mold that can be used to obtain a large number ofreplicas according to several possible techniques including casting on aUV-curable material, injection or compression molding, and reactive ionetching into a substrate.

[0176] A particularly preferred technique for forming the structuredscreens of the present invention is described in commonly assigned U.S.patent application Ser. No. 09/094,340, filed Jun. 9, 1998, and entitled“Method for Making Optical Micro-Structures Which Can Have ProfileHeights Exceeding 15 Microns,” which was published on Dec. 16, 1999 asPCT Patent Publication No. WO 99/64929, the contents of which in theirentireties are incorporated herein by reference. Using these techniques,microstructure shape accuracy and location to within, for example,fractions of a micron can be achieved. Moreover, by using thesetechniques to produce durable tools, the structured screens of thepresent invention can be produced inexpensively and in large volumes.

VI. EXAMPLES

[0177] The following, non-limiting, examples illustrate the design ofstructured screens using the techniques discussed above. In eachexample, the features of the screen surface or surfaces are controlledwith enough precision at each individual location so as to allow theaccurate shaping (control) of the light spreading (scattering) patternin the far field, which is the typical location for an observer. Inparticular, since the scattering surface is known in detail, it ispossible to reliably model its optical behavior using diffractionequations and thus determine the expected performance of the screen aswell as the tolerances involved in fabricating the screen.

[0178] As discussed above, to properly tailor the scattering pattern oneneeds to make use of several degrees of freedom, which can be basicallyseparated into local and global components. The screen itself iscomposed of micro-elements (microstructures) that act collectively togenerate the desired scattering. Each microstructure can be defined by aset of parameters. These are the local components. For example, in thecase of microlenses, the local parameters could be radii of curvature,conic constants, diameters, and so on. The global parameters define thelaws that must be obeyed by the local parameters (such as probabilitydistribution functions) and the spatial location of each microstructurewith respect to each other.

[0179] As also discussed above, to define the spatial arrangement andpositioning of the microstructures, one can employ a conveniently placedcoordinate system that defines the center of each microstructure and aglobal origin. Global components are defined with respect to thiscoordinate system. At the origin of each microstructure one can alsoassociate a local coordinate system that provides the referencenecessary to mathematically define the microstructure. It is through theuse of local and global components that one can shape to a virtuallyunlimited degree the scattering pattern produced by the screen.

[0180] The examples which follow illustrate the effects of variousglobal and local parameters on the scattering pattern of a screen. Inparticular, Example 1 illustrates the importance of local degrees offreedom in the shape of the scattering pattern, Examples 2 and 3 dealwith vertical offset and the general spatial placement ofmicrostructures on the screen surface or surfaces, respectively, andExample 4 addresses randomization of screen parameters. Finally,Examples 5 and 6 present illustrative applications of the invention tothe production of diffusing and display screens.

[0181] Prescriptions for the structured screens of the various examples,with reference to the figures to which the prescriptions correspond, areset forth the Screen Design Table which appears at the end of theexamples.

Example 1 Effect of Microstructure Shape

[0182] This example illustrates the importance of the shape of eachmicrostructure in tailoring the light spreading pattern.

[0183] Initially we consider a regular array of identicalmicrostructures with a shape that can be expressed as function of one ormore parameters, some of which are random variables. For instance, amicrostructure of spherical shape has a sag function that is given byR−(R²−r²)^(½), where R denotes the radius of curvature and r denotes theradial position from the origin. According to this definition, the arrayconsists of spherical microlenses with possibly variable radii ofcurvature. The size of each microstructure is identical throughout thescreen surface, so that the presence of randomness is confined to aparticular parameter, namely, the radii of curvature of the sphericalmicrolenses.

[0184] Consider initially a screen made of silicon to operate in theinfrared in a wavelength range from 2 to 4 microns. FIG. 26 shows thefar-field scattering profile for an array of spherical microlenses ofdiameter 100 microns. The microlens sag is fixed at 10 microns.

[0185] If we now replace the spherical microlenses by anothermicrostructure of a distinctively different shape, triangular forinstance, the scattering pattern shown in FIG. 27 is found. Note thateach microstructure is an isosceles triangle of base 100 microns anddepth 10 microns, i.e., the basic dimensions are the same as for thespherical array. Since the array is more akin to a triangular gratingone observes the two separated intensity peaks corresponding to the twomain diffraction orders.

[0186] As a further example, consider a hyperbolic profile with radiusof curvature R=120 microns and conic constant κ=−2. With theseparameters the microlens sag is again 10 microns for a diameter of 100microns. As shown in FIG. 28, although the general scattering profile issimilar to that observed with spherical microlenses (see FIG. 26), onenotes the increased peak intensity at the extremes of the pattern.

[0187] The effect of microstructure shape illustrated in FIG. 26 to FIG.28 can be even more dramatic as the numerical aperture increases.

[0188] As a further example of the effect of microstructure shape,consider a screen in acrylic designed to operate between 400 nm and 700nm. The screen is composed of a regular array of spherical microlensesof diameter 50 microns. FIG. 29 and FIG. 30 illustrate the scatteringprofile for such an array for two microlens depths, namely, 12 micronsin FIG. 29 and 20 microns in FIG. 30. As can be seen in these figures,microlens depth provides another parameter for shaping the scatteringpattern of a screen.

[0189] The case of a parabolic microlens shape with 20 microns total sag(total depth) is shown in FIG. 31. The parabolic profile allows thegeneration of scattering profiles of considerable flatness (and, as aresult, high gain) in comparison with spherical or any other profiles.In the low sag limit there is little distinction between spherical andparabolic profiles and the scattering is virtually identical. However,as the microlens sag increases the difference between these two profilesbecomes no longer negligible and this reflects directly on thescattering.

[0190] The important lesson to be learned here is that the shape of themicrostructure can be used to control the shape of the scatteringprofile. We have presented some examples that show the variability ofthe scattering as a function of the shape used. It is possible to takethe inverse path and define some scattering pattern of interest and, bymeans of optimization algorithms, calculate the sag profile that bestapproximates the desired scattering pattern.

[0191] As a final note we mention that, while the above examples havebeen confined to regular arrays, it is generally desirable to considerrandom arrays to avoid effects due to the periodicity of the array. Withrandomization, large intensity fluctuations tend to be minimized in thescattering pattern. Furthermore, it tends to eliminate undesirablevisual effects such as aliasing or moire fringes due to thesuperposition of two locally periodic patterns. Randomization, however,does not significantly alter the basic shape observed with the regulararray. It just makes it more homogeneous and robust.

Example 2 Effect of Vertical Offset

[0192] As illustrated in Example 1, the microstructure profile basicallydetermines the shape and the divergence span of the scattering pattern.Randomizing the profile helps reduce the high-frequency oscillations andattain a smoother pattern. Depending on the spectral band of operation,hot spots may occur that are detrimental to the performance of thescreen (see, for example, FIG. 26).

[0193] An important element useful in homogenizing the scatteringpattern and providing a further parameter to help decrease the influenceof hot spots is the vertical offset or piston of the microlens. Itseffect is basically to displace a microlens along its axis of symmetryby a definite quantity. The net result is that the relative position ofthe vertex or origin of a microstructure varies as a function of itsposition on the screen.

[0194] To illustrate the effect, we again consider the array used togenerate FIG. 26, that is, spherical microlenses of diameter 100 micronsand sag 10 microns. Cross-sections of the screen profile without andwith piston are shown in FIG. 32 and FIG. 33, respectively. The effectof the offset on the scattering pattern is shown in FIG. 34 which shouldbe compared with the pattern of FIG. 26. The disappearance of thecentral hot spot is evident in FIG. 34.

[0195] As a illustration of the homogenizing effect of a verticaloffset, consider the same array of spherical microlenses shown in FIG.33, but now operating in the visible, between 400 nm and 700 nm, insteadof the infrared range. The scattering patterns with and without thevertical offset are shown in FIG. 35 by the solid and dashed lines,respectively.

[0196] The above results make clear the effect of a vertical offset inreducing hot spots and homogenizing a scattering pattern. By using thiseffect alone or in combination with other randomizations, one canachieve smooth scattering patterns with a minimum of visual artifacts orstrong intensity fluctuations.

Example 3 Spatial Arrangement of Microstructures

[0197] A defining global parameter of great relevance to the scatteringpattern is the spatial arrangement of the microstructures on the screen.The main influence of the spatial arrangement is reflected in theoverall symmetry of the scattering pattern.

[0198] For instance, the regular square array of FIG. 36 generates athree-dimensional pattern that resembles a rectangle, depending on thedivergence angles introduced by the local configuration of themicrolenses. The hexagonal array of FIG. 37, on the other hand,generates a scattering pattern in the form of a hexagon which can bestretched along one direction more than another depending on thedivergence angles introduced by the configuration of the microlenses.

[0199] Apart from geometrical considerations, the overall symmetry isimportant in many aspects, but mainly because it affects how energy isconcentrated in the observation plane. For instance, for a pattern ofsimilar spatial extension, the hexagonal arrangement of FIG. 37concentrates more light than the square array of FIG. 36. As a result,the scattering pattern due to a hexagonal array exhibits more gain,i.e., scattering over a specified angular range, than an equivalentsquare array.

[0200] In addition to gain there are also important manufacturing issueswhich come into play when considering the implementation of a particularspatial arrangement. For example, because of the corners, square arraysare generally deeper than hexagonal arrays and thus more difficult tomanufacture.

[0201] From the perspective of the scattering pattern, the two factorsof primary relevance are the shape of the intensity profile and thedivergence angle. The shape is controlled by the profile of themicrostructures (sag function) while the divergence angle is controlledby the slope of the profile (first derivative of the sag function).

[0202] Therefore, it would seem that the particular size anddistribution of microstructures should not matter, as long as theindividual profiles are randomized for a smooth pattern properly definedto provide the desired divergence angles. In visual systems, however,there are additional effects associated with the spatial arrangement ofmicrostructures that are not apparent in the scattering pattern andreflect the interaction between the projected image and the samplingeffect caused by the existence of discrete individual microstructures.

[0203] A simple example of this interaction is aliasing, easily observedin pictures of high-frequency features taken with commercial digitalcameras. Another effect is the appearance of color bands that ariseagain as a result of finite sampling and the color distribution inprojected images. Independent of the screen performance in the farfield, unless the scattering elements are sufficiently small, effectsdue to sampling may arise and need to be addressed.

[0204] One way to eliminate these effects completely is to use a screendesign that renounces a regular spatial arrangement of microstructures.This can be done by employing a screen design where microstructures arecharacterized by both local parameters and a spatial boundary, where thespatial boundary is the closed line that surrounds the microstructure.For example, for a square array, the boundary is a square, while for ahoneycomb, the boundary is a hexagon.

[0205] In one dimension, a regular spatial arrangement can be avoided byusing cylinders of variable diameter. In two dimensions, the boundarycan be, in most general form, a polygonal curve. The complete set ofpolygonal curves that defines the boundaries of the microstructures inthe whole screen would generally (but not necessarily always) bearranged in close-packed form, as shown in FIG. 38. Another possiblearrangement, that also avoids a regular spatial arrangement and issimpler to implement, employs rectangular or other shaped cells tocompose a mosaic distribution of microlenses as shown in FIG. 39. Aparticular advantage of a mosaic, or a general polygonal boundary, isthat it reduces the influence of defects caused by the tiling of two ormore screen arrays to form a larger screen.

Example 4 Randomization Process

[0206] A particularly effective way to avoid the presence of artifactsdue to the periodic repetition of microstructures on a screen is to userandomization, which can be applied to either local parameters or globalparameters or to both local and global parameters, as desired.

[0207] Any set of random numbers satisfy a probability distributionfunction (PDF), which basically defines the probability of choosing agiven value in an allowed range of parameters. Because of itssimplicity, and availability in most computers, a preferred PDF is theuniform PDF. In this case, equal probability is assigned to each valuein the range of the parameter. With the ability to accurately shape eachindividual microstructure in the screen, the particular PDF to be usedin the randomization need not be confined to a few specific types butcan assume an arbitrary form. Furthermore, distinct parameters can besubjected to different PDF's, depending on the desired properties forthe scattering pattern.

[0208] As a simple illustration of the effect of a particular PDFassignment, consider the case of a random array of 100-micron diameterspherical microlenses with maximum sag in the range from 5 to 15 micronsand with a vertical offset equal to a randomized ±2 microns with auniform PDF.

[0209] The only available parameters for randomization for themicrolenses themselves are then the radius of curvature and the maximumsag of each microlens. However, in addition to the specific functionalshape of the PDF to be adopted, there are two distinct procedures toattain the randomization, depending on whether the PDF refers to theradius of curvature or the maximum lens sag. Since the sag is directlyrelated to the radius of curvature the PDF's will be likewise related.

[0210] An illustration of this effect is shown in FIG. 40 and FIG. 41.The result of changing the parameter to which the uniform PDF is appliedis shown in the scattering patterns of FIG. 42, where the pattern for auniform PDF in sag is represented by the solid line and that for auniform PDF in radius of curvature is represented by the dashed line. Ascan be seen in this figure, with the uniform distribution on sag, thescattering pattern tends to spread over a larger angular range. Thislarger angle scattering results from an increased number of deep-saglens units for the uniform distribution on sag.

[0211] Depending on the specifications of the scattering it may bedesirable to use other probability distribution functions besides auniform distribution. A Gaussian distribution, for instance, allows theconcentration of a parameter (e.g., sag or radius) around a givenaverage value. There is no limitation on the types of PDF's that can beutilized, the only restrictions being imposed by the desired performanceof the system. The uniform distribution tends to be the distribution ofchoice for most applications, mainly due to its simplicity. In manycases of interest this is sufficient to meet the system requirements.However, the option to implement an arbitrary PDF provides an importantdegree of freedom for tailoring the scattering pattern.

Example 5 Diffusing Screen

[0212] As discussed above, in general terms, screens can be divided intotwo categories, i.e., diffusing screens and display screens, based onthe different requirements involved in these two applications.

[0213] Although both types of screens are intended to spread theincident illumination homogeneously over an area larger than the area itwould have been spread over without the screen, display applicationstypically involve a number of additional requirements related to thevisual interaction with an external user. A diffusing screen, on theother hand, does not necessarily need to be used visually.

[0214] For instance, an optical detection system might require a larger,homogeneous focal spot for a given spectral band than would be providedwithout a screen. The absence of visual performance requirementstypically makes the design of diffusing screens somewhat easier sinceone must concentrate only on the size and shape of the diffusingpattern. Also, the required angular spread tends to be small, whichimplies the use of shallow microstructures which are easier tofabricate.

[0215] As an example, consider an input beam equal to 4 mm in thespectral range of 2 to 4 microns which is being focused at a distanceequal to 10 mm. With these parameters, the full divergence angle isapproximately 0.14 degrees. However, let us assume that a beam with fulldivergence at half-maximum of 26 degrees is required. This task can beaccomplished with the microlens array shown in FIG. 43. The resultingdiffraction pattern, averaged over the spectral range is shown in FIG.44 and cross-sections of the scattering pattern are shown on FIG. 45.

Example 6 Display Screen

[0216] As mentioned above, screens for display applications generallypresent a number of challenges in addition to spreading the illuminationinto a specific angular range with a certain scattering profile. Here wewill be concerned only with those requirements that pertain to theproperties of the scattering pattern. Visual requirements, e.g., color,contrast, and various subjective variables, involve issues that cannotbe easily modeled by diffraction calculations. These are betterevaluated by direct observation of a test image on a screen sample.

[0217] A first immediate distinction from diffusive screens is thatdisplay screens are confined to the visual part of the spectrum. Anothermain difference is that displays generally require a larger angularrange, typically spanning a total of 100 degrees useful field. Inaddition, the angular divergence may be different in the vertical andhorizontal directions.

[0218] An example of a scattering pattern for a display screen is shownin FIG. 46. The divergence along the fast axis is 36 degrees (estimatedgain 4.9) while along the slow axis it is about 17 degrees (estimatedgain 16.9). This pattern was generated with the hexagonal array ofanamorphic microlenses illustrated in FIG. 47.

[0219] Another example, obtained with an anamorphic mosaic array isshown in FIG. 48, with a portion of the array itself being shown in FIG.49. For this mosaic design, randomness was introduced for both theindividual microlens elements as well as for the sizes of therectangular boundaries (25 to 30 microns in one direction and 45 to 50microns in the other). The divergence along the fast axis is 33 degrees(estimated gain 5.5) while that along the slow axis is about 15 degrees(estimated gain 9.6).

[0220] Although specific embodiments of the invention have beendescribed and illustrated, it will be apparent to those skilled in theart that modifications and variations can be made without departing fromthe invention's spirit and scope. The following claims are thus intendedto cover the specific embodiments set forth herein as well as suchmodifications, variations, and equivalents. Screen Design Tables Arraytype Cylindrical Spatial arrangement Close-packed Microstructurediameter 100 μm Microstructure profile Spherical Microstructure sagrange 10-10 μm Total array depth range 10 μm Microstructurerandomization PDF Regular array Vertical offset 0 μm Offsetrandomization PDF Not applicable Wavelength range 2-4 μm Array typeCylindrical Spatial arrangement Close-packed Microstructure diameter 100μm Microstructure profile Triangular Microstructure sag range 10-10 μmTotal array depth range 10 μm Microstructure randomization PDF Regulararray Vertical offset 0 μm Offset randomization PDF Not applicableWavelength range 2-4 μm Array type Cylindrical Spatial arrangementClose-packed Microstructure diameter 100 μm Microstructure profileAspheric Microstructure sag range 10-10 μm Total array depth range 10 μmMicrostructure randomization PDF Regular array Vertical offset 0 μmOffset randomization PDF Not applicable Wavelength range 2-4 μm Arraytype Cylindrical Spatial arrangement Close-packed Microstructurediameter 50 μm Microstructure profile Spherical Microstructure sag range12-12 μm Total array depth range 12 μm Microstructure randomization PDFRegular array Vertical offset 0 μm Offset randomization PDF Notapplicable Wavelength range 0.4-0.7 μm Array type Cylindrical Spatialarrangement Close-packed Microstructure diameter 50 μm Microstructureprofile Spherical Microstructure sag range 20-20 μm Total array depthrange 20 μm Microstructure randomization PDF Regular array Verticaloffset 0 μm Offset randomization PDF Not applicable Wavelength range0.4-0.7 μm Array type Cylindrical Spatial arrangement Close-packedMicrostructure diameter 50 μm Microstructure profile ParabolicMicrostructure sag range 20-20 μm Total array depth range 20 μmMicrostructure randomization PDF Regular array Vertical offset 0 μmOffset randomization PDF Not applicable Wavelength range 0.4-0.7 μmArray type Cylindrical Spatial arrangement Close-packed Microstructurediameter 100 μm Microstructure profile Spherical Microstructure sagrange 10-10 μm Total array depth range 10 μm Microstructurerandomization PDF Regular array Vertical offset 0 μm Offsetrandomization PDF Not applicable Wavelength range 0.4-0.7 μm Array typeCylindrical Spatial arrangement Close-packed Microstructure diameter 100μm Microstructure profile Spherical Microstructure sag range 10-10 μmTotal array depth range 10 μm Microstructure randomization PDF Regulararray Vertical offset ±2 μm Offset randomization PDF Uniform Wavelengthrange 0.4-0.7 μm Array type Cylindrical Spatial arrangement Close-packedMicrostructure diameter 100 μm Microstructure profile SphericalMicrostructure sag range 10-10 μm Total array depth range 10 μmMicrostructure randomization PDF Regular array Vertical offset ±2 μmOffset randomization PDF Uniform Wavelength range 0.4-0.7 μm Array typeTwo-dimensional Spatial arrangement Square close-packed Microstructurediameter 100 μm Microstructure profile Spherical Microstructure sagrange 10-10 μm Total array depth range 10 μm Microstructurerandomization PDF Regular array Vertical offset 0 μm Offsetrandomization PDF Not applicable Wavelength range 0.4-0.7 μm Array typeTwo-dimensional Spatial arrangement Hexagonal Microstructure diameter100 μm Microstructure profile Spherical Microstructure sag range 10-10μm Total array depth range 10 μm Microstructure randomization PDFRegular array Vertical offset 0 μm Offset randomization PDF Notapplicable Wavelength range 0.4-0.7 μm Array type Two-dimensionalSpatial arrangement Random polygonal boundaries Microstructure diameter50 μm (Average) Microstructure profile Spherical Microstructure sagrange 2-10 μm Total array depth range 10 μm Microstructure randomizationPDF Uniform Vertical offset 0 μm Offset randomization PDF Not applicableWavelength range 0.4-0.7 μm Array type Two-dimensional Spatialarrangement Mosaic Microstructure diameter 50 μm (Average)Microstructure profile Spherical (Anamorphic) Microstructure sag range2-10 μm Total array depth range 10 μm Microstructure randomization PDFUniform Vertical offset 0 μm Offset randomization PDF Not applicableWavelength range 0.4-0.7 μm Array type Two-dimensional Spatialarrangement Hexagonal Microstructure diameter 500 μm (Average)Microstructure profile Spherical (Anamorphic) Microstructure sag range2-8 μm Total array depth range 14 μm Microstructure randomization PDFUniform Vertical offset ±2 μm Offset randomization PDF UniformWavelength range 0.4-0.7 μm Array type Two-dimensional Spatialarrangement Hexagonal Microstructure diameter 50 μm (Average)Microstructure profile Spherical (Anamorphic) Microstructure sag range2.5-4 μm (slow axis) 9-11 μm (fast axis) Total array depth range 16 μmMicrostructure randomization PDF Uniform Vertical offset ±2 μm Offsetrandomization PDF Uniform Wavelength range 0.4-0.7 μm Array typeTwo-dimensional Spatial arrangement Mosaic Microstructure diameter 25-30mm (slow axis) 45-50 mm (fast axis) Microstructure profile SphericalMicrostructure sag range 1.9-3.5 μm (slow axis) 8.8-10 μm (fast axis)Total array depth range 17.4 μm Microstructure randomization PDF UniformVertical offset ±2 μm Offset randomization PDF Uniform Wavelength range0.4-0.7 μm

What is claimed is:
 1. A method for making a structured screen thatprovides a desired spread of incident light, said structured screencomprising a substrate and a plurality of microstructures distributedover at least one surface of said substrate, said method comprising: (a)selecting a location on said at least one surface of the substrate foreach of said plurality of microstructures; (b) selecting a configurationfor each of said plurality of microstructures; (c) calculating thespread of the incident light for the selected locations and the selectedconfigurations of steps (a) and (b); (d) comparing the calculated spreadof step (c) with the desired spread and, if necessary, repeating atleast one of steps (a) and (b), and step (c) until the comparisonbetween the calculated spread and desired spread satisfies a specifiedcriterion; and (e) producing a plurality of microstructures having, toan accuracy of better than 10·λ_(n), the locations and theconfigurations which, in step (d), resulted in the satisfaction of thespecified criterion, where λ_(n) is the nominal operating wavelength forthe screen.
 2. The method of claim 1 wherein the locations selected instep (a) form a regular array.
 3. The method of claim 2 wherein thearray is a hexagonal array.
 4. The method of claim 1 wherein thelocations selected in step (a) are based on a set of unit cells whichform a mosaic.
 5. The method of claim 4 wherein the mosaic is random. 6.The method of claim 4 wherein the structured screen has internalmicrostructures and edge microstructures and the mosaic provides atleast some junctions between internal microstructures that correspond,in terms of light spreading, to at least some junctions between edgemicrostructures resulting from the tiling of two structured screens toone another.
 7. The method of claim 1 wherein at least some of thelocations selected in step (a) are randomly distributed in accordancewith a predetermined probability density function.
 8. The method ofclaim 1 wherein the locations of the microstructures are based on arandom set of polygonal shaped boundaries.
 9. The method of claim 1wherein in step (b) at least a portion of at least some of themicrostructures is selected to have a configuration given by:${s\left( {x,y} \right)} = {\frac{c\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}{1 + \sqrt{1 - {\left( {\kappa + 1} \right){c^{2}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}}}} + {\sum\limits_{p}{A_{p}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}^{p/2}}}$

where s(x,y) is the sag of said portion, c is its curvature, (x_(c),y_(c)) is its center point, κ is a conic constant, and A_(p) areaspheric coefficients.
 10. The method of claim 9 wherein A_(p)≠0 for atleast one p.
 11. The method of claim 9 wherein κ≠0.
 12. The method ofclaim 9 wherein: κ=−1; and A_(p)=0 for all p.
 13. The method of claim 1wherein in step (b) at least a portion of at least some of themicrostructures is selected to have a configuration given by:${s\left( {x,y} \right)} = {{\sum\limits_{p = 1}^{\infty}{B_{p}\left( {x - x_{c}} \right)}^{p}} + {C_{p}\left( {y - y_{c}} \right)}^{p}}$

where s(x,y) is the sag of said portion, (x_(c), y_(c)) is its centerpoint, and B_(p) and C_(p) are power series coefficients.
 14. The methodof claim 1 wherein at least some of the microstructures comprise (i) acurved, microlens portion and (ii) a straight-sided, piston portion. 15.The method of claim 1 wherein at least some of the microstructurescomprise an anamorphic microlens.
 16. The method of claim 1 wherein instep (b) at least a portion of at least some of the microstructures isselected to have a configuration given by:${s\left( {x,y} \right)} = \frac{{c_{x}\left( {x - x_{c}} \right)}^{2} + {c_{y}\left( {y - y_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{x}} \right){c_{x}\left( {x - x_{c}} \right)}^{2}} + {\left( {1 + \kappa_{y}} \right){c_{y}\left( {y - y_{c}} \right)}^{2}}}}$

where s(x,y) is the sag of said portion, (x_(c), y_(c)) is its centerpoint, c_(x) and c_(y) are curvatures along x and y, respectively, andκ_(x) and κ_(y) are conic constants along x and y, respectively.
 17. Themethod of claim 1 wherein in step (b) at least a portion of at leastsome of the microstructures is selected to have a configuration givenby:${s\left( {x,y} \right)} = {\frac{{c_{x}\left( {x - x_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{x}} \right)\left( {x - x_{c}} \right)^{2}}}} + \frac{{c_{y}\left( {y - y_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{y}} \right)\left( {y - y_{c}} \right)^{2}}}} + {\sum\limits_{p}{A_{xp}\left( {x - x_{c}} \right)}^{p}} + {A_{yp}\left( {y - y_{c}} \right)}^{p}}$

where s(x,y) is the sag of said portion, (x_(c), y_(c)) is its centerpoint, c_(x) and c_(y) are curvatures along x and y, respectively, κ_(x)and κ_(y) are conic constants along x and y, respectively, and A_(xp)and A_(yp) are aspheric coefficients along x and y, respectively. 18.The method of claim 1 wherein: (a) at least a portion of at least someof the microstructures is selected to have a configuration characterizedby at least one parameter; and (b) said at least one parameter israndomly distributed in accordance with a predetermined probabilitydensity function.
 19. The method of claim 18 wherein the at least onerandomly distributed parameter has a uniform probability densityfunction over a predetermined range for the parameter.
 20. The method ofclaim 18 wherein the at least one randomly distributed parameter isradius of curvature.
 21. The method of claim 18 wherein the at least onerandomly distributed parameter is maximum surface sag.
 22. The method ofclaim 18 wherein the at least one randomly distributed parameter ischaracteristic of the transverse size of a microstructure.
 23. Themethod of claim 22 wherein the parameter is diameter.
 24. The method ofclaim 1 wherein: (a) at least some of the microstructures comprise (i) acurved, microlens portion and (ii) a straight-sided, piston portion; and(b) the heights of the straight-sided, piston portions are randomlydistributed in accordance with a predetermined probability densityfunction.
 25. The method of claim 24 wherein the heights of thestraight-sided, piston portions have a uniform probability densityfunction over a predetermined range for said heights.
 26. The method ofclaim 1 wherein: (a) at least some of the microstructures have an apex,said apex being separated from the substrate by a distance; and (b) atleast some of said distances are randomly distributed in accordance witha predetermined probability density function.
 27. The method of claim 26wherein said randomly distributed distances have a maximum value and thedifference between said maximum value and said randomly distributeddistances has a uniform probability density function over apredetermined range for said difference.
 28. The method of claim 1wherein the substrate defines a first optical axis and the configurationof at least some of the microstructures comprises a microlens whichdefines a second optical axis which is not parallel to the first opticalaxis.
 29. The method of claim 1 wherein as produced in step (e), theplurality of microstructures have, to an accuracy of better than5·λ_(n), the locations and the configurations which, in step (d),resulted in the satisfaction of the specified criterion.
 30. The methodof claim 1 wherein step (e) comprises direct laser writing in aphotoresist.
 31. The method of claim 1 wherein microstructures aredistributed over two of the substrate's surfaces.
 32. Apparatus forcontrolled spreading of light comprising a plurality of microstructures,each microstructure being located with better than 10·λ_(n) accuracy ata predetermined location with respect to all other microstructures andeach microstructure having a configuration that corresponds, with betterthan 10·λ_(n) accuracy, to a predetermined mathematical relation, whereλ_(n) is the nominal operating wavelength of the apparatus and saidpredetermined locations and predetermined mathematical relations allowan a priori calculation of the spreading of incident light by theapparatus.
 33. The apparatus of claim 32 wherein the predeterminedlocations form a regular array.
 34. The apparatus of claim 33 whereinthe array is a hexagonal array.
 35. The apparatus of claim 32 whereinthe predetermined locations are based on a set of unit cells which forma mosaic.
 36. The apparatus of claim 35 wherein the mosaic is random.37. The apparatus of claim 35 wherein the apparatus has internalmicrostructures and edge microstructures and the mosaic provides atleast some junctions between internal microstructures that correspond,in terms of light spreading, to at least some junctions between edgemicrostructures resulting from the tiling of two samples of theapparatus to one another.
 38. The apparatus of claim 32 wherein at leastsome of the predetermined locations are randomly distributed inaccordance with a predetermined probability density function.
 39. Theapparatus of claim 32 wherein the predetermined locations are based on arandom set of polygonal shaped boundaries.
 40. The apparatus of claim 32wherein at least a portion of the configuration of at least some of themicrostructures corresponds with better than 10·λ_(n) accuracy to themathematical relation:${s\left( {x,y} \right)} = {\frac{c\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}{1 + \sqrt{1 - {\left( {\kappa + 1} \right){c^{2}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}}}} + {\sum\limits_{p}{A_{p}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}^{p/2}}}$

where s(x,y) is the sag of said portion, c is its curvature, (x_(c),y_(c)) is its center point, κ is a conic constant, and A_(p) areaspheric coefficients.
 41. The apparatus of claim 40 wherein A_(p)≠0 forat least one p.
 42. The apparatus of claim 40 wherein κ≠0.
 43. Theapparatus of claim 40 wherein: (a) κ=−1; and (b) A_(p)=0 for all p. 44.The apparatus of claim 32 wherein at least a portion of theconfiguration of at least some of the microstructures corresponds withbetter than 10·λ_(n) accuracy to the mathematical relation:${s\left( {x,y} \right)} = {{\sum\limits_{p = 1}^{\infty}{B_{p}\left( {x - x_{c}} \right)}^{p}} + {C_{p}\left( {y - y_{c}} \right)}^{p}}$

where s(x,y) is the sag of said portion, (x_(c), y_(c)) is its centerpoint, and B_(p) and C_(p) are power series coefficients.
 45. Theapparatus of claim 32 wherein at least some of the microstructurescomprise (i) a curved, microlens portion and (ii) a straight-sided,piston portion.
 46. The apparatus of claim 32 wherein at least some ofthe microstructures comprise an anamorphic microlens.
 47. The apparatusof claim 32 wherein at least a portion of the configuration of at leastsome of the microstructures corresponds with better than 10 ·λ_(n)accuracy to the mathematical relation:${s\left( {x,y} \right)} = \frac{{c_{x}\left( {x - x_{c}} \right)}^{2} + {c_{y}\left( {y - y_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{x}} \right){c_{x}\left( {x - x_{c}} \right)}^{2}} + {\left( {1 + \kappa_{y}} \right){c_{y}\left( {y - y_{c}} \right)}^{2}}}}$

where s(x,y) is the sag of said portion, (x_(c), y_(c)) is its centerpoint, c_(x) and c_(y) are curvatures along x and y, respectively, andκ_(x) and κ_(y) are conic constants along x and y, respectively.
 48. Theapparatus of claim 32 wherein at least a portion of the configuration ofat least some of the microstructures corresponds with better than10·λ_(n) accuracy to the mathematical relation:${s\left( {x,y} \right)} = {\frac{{c_{x}\left( {x - x_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{x}} \right)\left( {x - x_{c}} \right)^{2}}}} + \frac{{c_{y}\left( {y - y_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{y}} \right)\left( {y - y_{c}} \right)^{2}}}} + {\sum\limits_{p}{A_{xp}\left( {x - x_{c}} \right)}^{p}} + {A_{yp}\left( {y - y_{c}} \right)}^{p}}$

where s(x,y) is the sag of said portion, (x_(c), y_(c)) is its centerpoint, c_(x) and c_(y) are curvatures along x and y, respectively, κ_(x)and κ_(y) are conic constants along x and y, respectively, and A_(xp)and A_(yp) are aspheric coefficients along x and y, respectively. 49.The apparatus of claim 32 wherein: (a) at least some of thepredetermined mathematical relations include at least one commonparameter; and (b) said at least one common parameter is randomlydistributed in accordance with a predetermined probability densityfunction.
 50. The apparatus of claim 49 wherein the at least onerandomly distributed common parameter has a uniform probability densityfunction over a predetermined range for said common parameter.
 51. Theapparatus of claim 49 wherein the at least one randomly distributedcommon parameter is radius of curvature.
 52. The apparatus of claim 49wherein the at least one randomly distributed common parameter ismaximum surface sag.
 53. The apparatus of claim 49 wherein the at leastone randomly distributed common parameter is a parameter characteristicof the transverse size of a microstructure.
 54. The apparatus of claim53 wherein the parameter is diameter.
 55. The apparatus of claim 32wherein: (a) at least some of the microstructures comprise (i) a curved,microlens portion, and (ii) a straight-sided, piston portion; and (b)the heights of the straight-sided, piston portions are randomlydistributed in accordance with a predetermined probability densityfunction.
 56. The apparatus of claim 55 wherein the heights of thestraight-sided, piston portions have a uniform probability densityfunction over a predetermined range for said heights.
 57. The apparatusof claim 32 wherein: (a) at least some of the microstructures have anapex; and (b) the heights of at least some of said apexes are randomlydistributed in accordance with a predetermined probability densityfunction.
 58. The apparatus of claim 57 wherein said randomlydistributed heights have a maximum value and the difference between saidmaximum value and said randomly distributed heights has a uniformprobability density function over a predetermined range for saiddifference.
 59. The apparatus of claim 32 wherein the apparatus definesa first optical axis and the configuration of at least some of themicrostructures comprises a microlens which defines a second opticalaxis which is not parallel to the first optical axis.
 60. The apparatusof claim 32 wherein each microstructure is located with better than5·λ_(n) accuracy at a predetermined location with respect to all othermicrostructures and each microstructure has a configuration that withbetter than 5·λ_(n) accuracy corresponds to a predetermined mathematicalrelation.
 61. The apparatus of claim 32 wherein the apparatus comprisestwo spaced-apart surfaces and the plurality of microstructures isdistributed over both said surfaces.
 62. The apparatus of claim 32wherein: (a) the apparatus comprises two spaced-apart surfaces, (b) theplurality of microstructures is distributed over one of said surfaces;and (c) the other surface is a Fresnel lens.
 63. A microstructure foruse in an optical device comprising (i) a curved, microlens portion and(ii) a straight-sided, piston portion.
 64. The microstructure of claim63 wherein the curved, microlens portion has a spherical shape.
 65. Themicrostructure of claim 63 wherein the curved, microlens portion has aparabolic shape.
 66. Apparatus for controlled spreading of lightcomprising a plurality of microstructures wherein at least a portion ofeach microstructure is described by the equation:${s\left( {x,y} \right)} = {\frac{c\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}{1 + \sqrt{1 - {\left( {\kappa + 1} \right){c^{2}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}}}} + {\sum\limits_{p}{A_{p}\left\lbrack {\left( {x - x_{c}} \right)^{2} + \left( {y - y_{c}} \right)^{2}} \right\rbrack}^{p/2}}}$

where s(x,y) is the sag of said portion, c is a predetermined curvature,(x_(c), y_(c)) is a predetermined center point, κ is a predeterminedconic constant, A_(p) are predetermined aspheric coefficients, and atleast κ or one of the A_(p)'s is not equal to zero.
 67. The apparatus ofclaim 66 wherein: (a) κ=−1; and (b) A_(p)=0 for all p.
 68. Apparatus forcontrolled spreading of light comprising a plurality of microstructureswherein at least a portion of each microstructure is described by theequation:${s\left( {x,y} \right)} = \frac{{c_{x}\left( {x - x_{c}} \right)}^{2} + {c_{y}\left( {y - y_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{x}} \right){c_{x}\left( {x - x_{c}} \right)}^{2}} + {\left( {1 + \kappa_{y}} \right){c_{y}\left( {y - y_{c}} \right)}^{2}}}}$

where s(x,y) is the sag of said portion, (x_(c), y_(c)) is apredetermined center point, c_(x) and c_(y) are predetermined, non-equalcurvatures along x and y, respectively, and KX and Ky are predeterminedconic constants along x and y, respectively.
 69. Apparatus forcontrolled spreading of light comprising a plurality of microstructureswherein at least a portion of each microstructure is described by theequation:${s\left( {x,y} \right)} = {\frac{{c_{x}\left( {x - x_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{x}} \right)\left( {x - x_{c}} \right)^{2}}}} + \frac{{c_{y}\left( {y - y_{c}} \right)}^{2}}{1 + \sqrt{1 - {\left( {1 + \kappa_{y}} \right)\left( {y - y_{c}} \right)^{2}}}} + {\sum\limits_{p}{A_{xp}\left( {x - x_{c}} \right)}^{p}} + {A_{yp}\left( {y - y_{c}} \right)}^{p}}$

where s(x,y) is the sag of said portion, (x_(c), y_(c)) is apredetermined center point, c_(x) and c_(y) are predetermined, non-equalcurvatures along x and y, respectively, κ_(x) and κ_(y) arepredetermined conic constants along x and y, respectively, and A_(xp)and A_(yp) are predetermined aspheric coefficients along x and y,respectively.
 70. Apparatus for controlled spreading of light comprisinga plurality of microstructures, each microstructure having aconfiguration that is characterized by at least one predeterminedparameter which is randomly distributed in accordance with apredetermined probability density function.
 71. The apparatus of claim70 wherein the at least one randomly- distributed parameter has auniform probability density function.
 72. The apparatus of claim 70wherein: (a) each microstructure comprises (i) a curved, microlensportion and (ii) a straight-sided, piston portion; and (b) therandomly-distributed parameter characterizes the straight-sided, pistonportion.
 73. The apparatus of claim 70 wherein each microstructure ischaracterized by two predetermined parameters, each of which is randomlydistributed in accordance with a predetermined probability densityfunction which may be the same or different for the two parameters. 74.The apparatus of claim 73 wherein each of the randomly-distributedparameters has a uniform probability density function over apredetermined range for the parameter.
 75. The apparatus of claim 73wherein: (a) each microstructure comprises (i) a curved, microlensportion and (ii) a straight-sided, piston portion; and (b) one of thetwo randomly-distributed parameters characterizes the curved, microlensportion and the other randomly- distributed parameter characterizes thestraight-sided, piston portion.
 76. The apparatus of claim 70 whereinthe locations of the microstructures is randomized in accordance with apredetermined probability density function.
 77. A structured screencomprising a plurality of predetermined microstructures, wherein: (a)said microstructures comprise (i) a curved, microlens portion and (ii) astraight-sided, piston portion which has a predetermined height whichcan be zero: (b) said curved, microlens portions have predetermineddiameters and predetermined maximum sags; and (c) for at least some ofsaid microlenses, the sum of the predetermined maximum sag and thepredetermined height is greater than the predetermined diameter.
 78. Thestructured screen of claim 77 wherein at least one of the predetermineddiameters, the predetermined maximum sags, and the predetermined heightsis randomly distributed in accordance with a predetermined probabilitydensity function.
 79. The structured screen of claim 78 wherein thepredetermined diameters have a uniform probability density function overa predetermined range for said diameters.
 80. The structured screen ofclaim 78 wherein the predetermined maximum sags have a uniformprobability density function over a predetermined range for said maximumsags.
 81. The structured screen of claim 78 wherein the predeterminedheights have a uniform probability density function over a predeterminedrange for said heights.
 82. A structured screen comprising a pluralityof predetermined aspherical microlenses, wherein said microlenses: (a)have predetermined diameters and predetermined maximum sags; and (b)produce a spread of incident light which has a flatter intensitydistribution than that produced by a plurality of spherical microlenseshaving the same predetermined diameters and predetermined sags.
 83. Thestructured screen of claim 82 wherein at least one of the predetermineddiameters and the predetermined maximum sags is randomly distributed inaccordance with a predetermined probability density function.
 84. Thestructured screen of claim 83 wherein the predetermined diameters have auniform probability density function over a predetermined range for saiddiameters.
 85. The structured screen of claim 83 wherein thepredetermined maximum sags have a uniform probability density functionover a predetermined range for said maximum sags.
 86. The structuredscreen of claim 82 wherein at least some of the microlenses areparabolic.
 87. A structured screen which defines an optical axis andcomprises a plurality of microstructures at least some of which comprisea microlens having an optical axis which is not parallel to the opticalaxis of the structured screen.
 88. A structured screen comprising: (a) aFresnel lens which comprises a plurality of surfaces in the form ofconcentric rings; and (b) a plurality of microstructures distributedover at least some of said plurality of surfaces, said plurality ofmicrostructures serving to control the spread of light incident on thestructured screen.
 89. A structured screen comprising a plurality ofunit cells and a plurality of microstructures, one microstructureassociated with each unit cell, wherein the perimeters of the unit cellsare non-regular polygons.
 90. The structured screen of claim 89 whereinthe perimeters are defined by a predetermined probability densityfunction.
 91. A structured screen comprising a plurality ofmicrostructures at least some of which comprise a microlens having afirst curvature in a first direction and a second curvature in a seconddirection orthogonal to the first direction, at least one of said firstand second curvatures being randomly distributed in accordance with apredetermined probability density function.
 92. The structured screen ofclaim 91 where both the first and second curvatures are randomlydistributed in accordance with a predetermined probability densityfunction which may be the same or different for the two curvatures. 93.A structured screen comprising: (a) a first sub-screen comprising aplurality of internal microstructures and a plurality of edgemicrostructures, each microstructure being located at a predeterminedlocation with respect to all other microstructures, said predeterminedlocations being based on a first set of unit cells which form a firstmosaic; (b) a second sub-screen comprising a plurality of internalmicrostructures and a plurality of edge microstructures, eachmicrostructure being located at a predetermined location with respect toall other microstructures, said predetermined locations being based on asecond set of unit cells which form a second mosaic; wherein: (i) thefirst and second sub-screens are tiled to one another, said tilingproducing edge junctions between edge microstructures of the firstsub-screen and edge microstructures of the second sub-screen; and (ii)each of the first and second mosaics provides at least some internaljunctions between internal microstructures that correspond, in terms oflight spreading, to at least some of the edge junctions.
 94. Thestructured screen of claim 93 wherein each of the first and secondmosaics is random.
 95. The structured screen of claim 93 wherein thefirst and second sub- screens are identical.